Weighted Recognizability over Infinite Alphabets Maria Pittou

Weighted Recognizability over Infinite Alphabets

Maria Pittou

and George Rahonis

Dedicated to the memory of Zolt´an ´Esik Abstract

We introduce weighted variable automata over infinite alphabets and com- mutative semirings. We prove that the class of their behaviors is closed under sum, and under scalar, Hadamard, Cauchy, and shuffle products, as well as star operation. Furthermore, we consider rational series over infinite alpha- bets and we state a Kleene-Sch¨utzenberger theorem. We introduce a weighted monadic second order logic and a weighted linear dynamic logic over infinite alphabets and investigate their relation to weighted variable automata. An application of our theory, to series over the Boolean semiring, concludes to new results for the class of languages accepted by variable automata.

Keywords: infinite alphabets, semirings, weighted variable automata, weighted MSO, weightedLDL

1 Introduction

The last two decades a large body of research has been devoted to the develop- ment of models for infinite state systems which have finite control structure and handle data from an unbounded domain. This research led to the concept of finite automata over infinite alphabets. Motivating examples for such models consist, for instance, XML schemas, software with integer parameters, and system specification and verification. Later on, it came up that finite automata over infinite alphabets can contribute also to a series of interesting topics namely, the problem of query graph databases [33], reasoning about systems with resource generation capabilities [10, 11], learning theories [30], and systems with freshness needed in object-oriented languages and security protocols [6].

Several models of automata with data values, i.e., over infinite alphabets have been investigated, namely register [24, 28, 29, 37], data [5], pebble [28, 36, 39], nominal [10], variable [21, 22], and P automata [9]. All these models refer to qualitative aspects of infinite state systems. Furthermore, rational [1, 25] and logic definable languages [4, 36] have been studied over infinite alphabets.

aDepartment of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece.

E-mail:{mpittou,grahonis}@math.auth.gr

DOI: 10.14232/actacyb.23.1.2017.16

In this paper we intend to study automata models over infinite alphabets in the quantitative setup. Our motivation origins from the fact that several applications require a quantitative analysis of systems, for instance the resource usage control where resource variables are mapped to infinite domains [10, 11]. It is well-known that weighted automata is a reasonable tool for the description of quantitative fea- tures of computing systems [14]. According to our best knowledge, a quantitative counterpart for automata over infinite alphabets does not exist. In [8] the authors considered quantitative infinite alphabets to model controlled variables for the con- troller synthesis problem from incompatible situations. For our investigation, we chose the concept of variable automata from [21, 22]. Variable automata are sim- ple in their definition and implementation in contrast to other proposed models.

Despite their simplicity, variable automata and their extensions appeared to be expressive enough for several applications. Indeed, in [2] the authors introduced fresh variable automata to describe web services in which the agents exchange data ranging over infinite domains. Furthermore, in [3], fresh variable automata were equipped with guards consisting of equalities and disequalities. In [10] variable automata were extended to consume data words, in order to express security poli- cies (safety properties) for model checking programs that dynamically generate and operate over resources. Very recently, variable automata have been also used for querying graph databases [43]. In a similar approach, a variableLTLwas has been investigated in [23]. More precisely, the atomic propositions in that logic were pa- rameterized with variables over some finite or infinite domain in order to express specifications over large, possibly infinite domains. The model checking problem has been also studied for that setting (cf. also [38]).

We consider our weighted variable automata over an infinite alphabet Σ and a commutative semiring K, and provide a systematic study of the class of their behaviors. Our framework builds upon the techniques which were developed in [26, 27] for variable tree automata over infinite ranked alphabets. We prove that, if in addition the semiringK is idempotent, then the class of series accepted by our models is closed under sum, and scalar, Hadamard, Cauchy, and shuffle products, as well as under star operation. As we indicate by a simple example, the proofs for the aforementioned properties require new techniques than the well-known ones for recognizable series [14]. We define rational series over infinite alphabets and state a Kleene-Sch¨utzenberger type theorem. Furthermore, we introduce a weighted monadic second order logic and a weighted linear dynamic logic over infinite al- phabets. We show the expressive equivalence of the latter logic to our weighed au- tomata, whereas the corresponding equivalence requires fragments on the weighted monadic second order logic. Therefore, several well-known results from classical weighted automata theory hold also for our weighted automata over infinite alpha- bets. Moreover, by considering the Boolean semiringB, we derive as an application of our theory new results for the class of variable automata of [21, 22]. This shows the robustness of our theory and the theory of variable automata [21, 22].

Apart from this Introduction, the paper contains 7 sections. In Section 2 we present some preliminary background. In Section 3 we introduce our weighted variable automata and in Section 4 we establish the closure properties of the class

of series accepted by our models. Then, in Section 5 we consider rational series over infinite alphabets and state our Kleene-Sch¨utzenberger theorem. Sections 6 and 7, respectively are devoted to weighted monadic second order logic and weighted linear dynamic logic, and their relation to weighted variable automata. In Section 8 we expose the new results on variable automata derived by our theory. Finally, in the Conclusion, we present some ideas for future research.

A preliminary version of this paper appeared in [32] (cf. also [31]).

2 Preliminaries

Let Σ be an alphabet, i.e., a nonempty (potentially infinite) set. As usually, we denote by Σ^∗ the set of all finite words over Σ and Σ⁺= Σ^∗\ {ε}, whereεis the empty word. A subsetL⊆Σ^∗is a language over Σ.A wordw=σ0. . . σn−1, where σ₀, . . . , σ_n−1∈Σ (n≥1), is written also asw=w(0). . . w(n−1) where w(i) =σ_i for every 0 ≤i ≤n−1. For every finite word w = w(0). . . w(n−1) and every 0≤i≤n−1 we denote byw_≥ithe suffixw(i). . . w(n−1). IfSis a set, thenP(S) will denote the powerset ofS, and the notationS⁰ ⊆_{f in}Smeans thatS⁰ is a finite subset ofS.

Asemiring (K,+,·,0,1) is an algebraic structure such that (K,+,0) is a com- mutative monoid, (K,·,1) is a monoid, 06= 1,· is both left- and right-distributive over +, and 0·k = k·0 = 0 for every k ∈ K. If no confusion arises, we shall denote the semiring simply byKand the·operation simply by concatenation. The semiringKis calledcommutative if the monoid (K,·,1) is commutative. Moreover, K is calledadditively idempotent (or simply idempotent), if k+k =k for every k∈K. Finally,K is calledlocally finite if every finitely generated subsemiring is finite. Interesting examples of semirings are the following:

- the semiring (N,+,·,0,1) of natural numbers, - theBoolean semiring B= ({0,1},+,·,0,1),

- thetropical ormin-plus semiring (R+∪ {∞},min,+,∞,0) whereR+={r∈ R|r≥0},

- thearctical ormax-plus semiring (R+∪ {−∞},max,+,−∞,0), - theViterbi semiring ( [0,1],max,·,0,1),

- every bounded distributive lattice with the operations supremum and infi- mum, and especially thefuzzy semiring F = ([0,1],max,min,0,1).

All the above semirings, except the first one, are idempotent.

Let Σ be an alphabet and K a semiring. A formal series (or simply series) over Σ and K is a mapping s : Σ^∗ → K. For every w ∈ Σ^∗ we write (s, w) for the value s(w) and refer to it as the coefficient of s on w. The support of s is

the set supp(s) = {w ∈ Σ^∗ | (s, w) 6= 0}. A series with finite support is called a polynomial. The constant series ek (k ∈ K) is defined, for every w ∈ Σ^∗, by

ek, w

=k. Moreover, for every w ∈Σ^∗, we denote by wthe series determined, for everyu∈Σ^∗, by (w, u) = 1 if u=w and 0, otherwise. The class of all series over Σ andK is denoted as usual byKhhΣ^∗ii, and the class of polynomials over Σ andK byKhΣ^∗i. The characteristic series 1L∈KhhΣ^∗iiof a languageL⊆Σ^∗ is defined by (1L, w) = 1 ifw∈Land (1L, w) = 0 otherwise.

Let s, r ∈ KhhΣ^∗ii and k ∈ K. The sum s+r, the scalar products ks and sk as well as the Hadamard product sr are defined elementwise by (s+ r, w) = (s, w) + (r, w), (ks, w) = k·(s, w), (sk, w) = (s, w)·k, and (sr, w) = (s, w)·(r, w), respectively, for everyw ∈Σ^∗. It is well-known that the structures

KhhΣ^∗ii,+,,e0,e1 and

KhΣ^∗i,+,,e0,e1

are semirings, which moreover are commutative (resp. idempotent) wheneverKis commutative (resp. idempotent).

TheCauchy product of r and sis the seriesr·s∈KhhΣ^∗iidefined for every w∈Σ^∗ by

(r·s, w) = X

u,v∈Σ^∗ w=uv

((r, u)·(s, v)).

The nth-iteration rⁿ ∈ KhhΣ^∗ii (n ≥ 0) of a series r ∈ KhhΣ^∗ii is defined inductively by

r⁰=ε and rⁿ⁺¹=r·rⁿ forn≥0.

Then, we have (rⁿ, w) = X

u₁,...,u_n∈Σ^∗ w=u1...un

((r, u₁)·. . .·(r, u_n)) for every w ∈Σ^∗. A series r ∈ KhhΣ^∗iiis called proper whenever (r, ε) = 0. If r is proper, then for everyw∈Σ^∗ andn >|w|we have (rⁿ, w) = 0. Thestar r^∗∈KhhΣ^∗iiof a proper series r ∈ KhhΣ^∗ii is defined by r^∗ = P

n≥0

rⁿ. Thus, for every w ∈ Σ^∗ we have (r^∗, w) = X

0≤n≤|w|

(rⁿ, w).

Finally, theshuffle product of rand sis the seriesrs∈KhhΣ^∗iidefined for everyw∈Σ^∗ by

(rs, w) = X

u,v∈Σ^∗ w∈uv

((r, u)·(s, v))

whereuvdenotes the shuffle product ofuandv.

Next we turn to weighted automata. For this we assume the alphabet Σ to be finite. A weighted automaton over Σ and K is a quadruple A = (Q, in, wt, ter) whereQis thefinite state set, in:Q→Kis theinitial distribution,wt:Q×Σ× Q→ K is a mapping assigning weights to the transitions of the automaton, and ter:Q→K is thefinal (orterminal)distribution.

Letw=w(0). . . w(n−1)∈Σ^∗. Apath of Aover wis a sequence of transitions

Pw:= ((qi, w(i), qi+1))_{0≤i≤n−1}. Theweight ofPwis given by the value weight(Pw) =in(q0)· Y

0≤i≤n−1

wt((qi, w(i), qi+1))·ter(qn).

Thebehavior of Ais the serieskAk: Σ^∗→K whose coefficients are given by (kAk, w) =X

P_w

weight(Pw) for everyw∈Σ^∗.

A series s ∈ KhhΣ^∗ii is called recognizable if s = kAk for some weighted automatonA over Σ and K. As usual we denote by Rec(K,Σ) the class of rec- ognizable series over Σ and K. Two weighted automata A= (Q, in, wt, ter) and A⁰= (Q⁰, in⁰, wt⁰, ter⁰) over Σ andK are calledequivalent ifkAk=kA⁰k.

Finally, a weighted automaton A = (Q, in, wt, ter) over Σ and K is called normalized if there exist two statesqin, qter∈Q,qin6=qter, such that:

- in(q) = 1 ifq=qin, andin(q) = 0 otherwise, - ter(q) = 1 ifq=qter, andter(q) = 0 otherwise, and - wt((q, σ, q_in)) =wt((q_ter, σ, q)) = 0

for every q ∈ Q, σ ∈ Σ. We shall denote a normalized weighted automaton A= (Q, in, wt, ter) simply by A = (Q, qin, wt, qter). The next result has been proved by several authors, cf. for instance [18].

Proposition 1. Let A = (Q, in, wt, ter) be a weighted automaton over Σ and K. We can effectively construct a normalized weighted automaton A⁰ such that (kA⁰k, w) = (kAk, w)for every w∈Σ⁺ and(kA⁰k, ε) = 0.

3 Weighted variable automata

In this section we introduce the notion of our weighted variable automata. We show that the well-known constructions on weighted automata are not sufficient to obtain the closure properties of the class of series recognized by our models.

Therefore, we provide some supplementary matter and we state Lemma 1 which will be needed in the sequel in our constructions.

Let Σ, Σ⁰ be (infinite) alphabets. A relabeling from Σ to Σ⁰ is a mapping h: Σ → P(Σ⁰). Next let Γ ⊆f in Σ, Z be a finite set whose elements are called bounded variables andyan element which is called afree variable. We assume that the sets Σ,Z, and{y} are pairwise disjoint. A relabelinghfrom Γ∪Z∪ {y}to Σ is calledvalid if

(i) it is the identity on Γ,¹

1Abusing notation we identify{σ}withσ, for everyσ∈Γ.

(ii) card(h(z)) = 1 for every z∈Z,

(iii) his injective onZ and Γ∩h(Z) =∅, and (iv) h(y) = Σ\(Γ∪h(Z)).

The above definition means that the application ofhon a wordwover Γ∪Z∪ {y}

assigns to every occurrence of a symbolz ∈Z in wthe same symbol from Σ, but it is possible to assign different symbols from Σ to different occurrences ofy in w.

This justifies the names bounded and free for the set of variablesZand the variable y, respectively. It should be clear that a valid relabeling from Γ∪Z∪ {y} to Σ is well-defined if it is defined only on Z satisfying conditions (ii) and (iii). We shall denote by V R(Γ∪Z∪ {y},Σ) the set of all valid relabelings from Γ∪Z∪ {y} to Σ, and simply byV R(Γ∪Z∪ {y}) if the alphabet Σ is understood.

We set ∆ = Γ∪Z∪ {y}and letw∈Σ^∗. Thepreimage of wover ∆ is the set preim∆(w) ={u∈∆^∗| there existsh∈V R(∆) such thatu∈h⁻¹(w)}.

Now we are ready to introduce our weighted variable automata over the infinite alphabet Σ and a semiringK.

Definition 1. A weighted variable automaton (wva for short) over Σ andK is a pair A = hΣ, Ai where Σ is an infinite alphabet and A = (Q, in, wt, ter) is a weighted automaton over Γ_A and K. The input alphabet Γ_A of A is defined by ΓA= ΣA∪Z∪ {y}, where ΣA⊆f inΣ,Z is a finite alphabet of bounded variables, andy is a free variable.

Thebehavior of Ais the serieskAk: Σ^∗→Kwhose coefficients are determined by

(kAk, w) = X

u∈preim_ΓA(w)

(kAk, u)

for everyw∈Σ^∗. Clearly, the above sum is finite and thus (kAk, w) is well-defined for everyw∈Σ^∗.

Two wvaAandA⁰ over Σ andK are calledequivalent wheneverkAk=kA⁰k.

A seriesrover Σ andKis calledv-recognizableif there exists a wvaAsuch that r =kAk. We shall denote byV Rec(K,Σ) the class of v-recognizable series over Σ andK. It should be clear that every weighted automatonAover a subalphabet Σ⁰⊆f inΣ and K can be considered as a wva such that its transitions labelled by variables carry the weight 0. Therefore, we get the next result, where the strictness of the inclusion trivially holds by the definition of wva.

Proposition 2. S

Σ⁰⊆_{f in}Σ

Rec(K,Σ⁰)(V Rec(K,Σ).

Throughout the paper Σ will denote an infinite alphabet,Z a finite set of bounded variables,ya free variable, and Ka commutative semir- ing. In addition, in the present and the next section,Kwill be assumed to be idempotent.

In the sequel, we will call a wva A=hΣ, Aiover Σ andK, simply a wva.

Definition 2. A wvaA=hΣ, Aiis called normalized ifA is normalized.

Proposition 3. LetA=hΣ, Aibe a wva. We can effectively construct a normalized wva A⁰ such that (kA⁰k, w) = (kAk, w) for everyw∈Σ⁺ and(kA⁰k, ε) = 0.

Proof. We immediately obtain our result by Proposition 1 and Definition 2.

In the sequel, we wish to investigate closure properties of the classV Rec(K,Σ).

For this, we cannot apply the well-known constructions from classical weighted au- tomata theory. For instance, let A=hΣ, Ai be a normalized wva, where A = ({qin, q, q_ter}, qin, wt_A, q_ter), Γ_A = {a} ∪ {z} ∪ {y} and transitions with non-zero weights given bywt_A((q_in, a, q)) =wt_A((q, z, q_ter)) = 1. Consider also the normal- ized wvaA⁰=hΣ, A⁰iwhereA⁰= ({q⁰_in, q⁰_ter}, q_in⁰ , wt_A⁰, q⁰_ter), Γ_A⁰ ={a⁰}∪{z⁰}∪{y⁰} and wt_A⁰((q⁰_in, a⁰, q_ter⁰ )) = wt_A⁰((q⁰_in, y⁰, q_ter⁰ )) = 1. Moreover, let us assume that a6=a⁰. Clearly, (kAk, aa⁰) = 1 and (kA⁰k, a⁰) = 1. Nevertheless, if we consider the disjoint union ofA andA⁰, say the weighted automaton B, then a, a⁰ ∈ΓB which implies that we cannot apply a valid relabeling assigning the lettera⁰ toz. This in turn, implies that the wordaa⁰ does not belong to the support of the wva derived by the weighted automatonB. Furthermore, another problem of this construction is the choice of the free variable among y and y⁰ which moreover causes new in- consistencies. Similar, even more complex, situations arise for the constructions of wva proving closure under further properties like Hadamard, Cauchy, and shuffle product. Therefore, we state Lemma 1 below which will be of great importance to our constructions for the closure properties of the classV Rec(K,Σ). We shall need some preliminary matter.

Let A=hΣ, Ai be a wva where A= (Q, in, wt, ter) with Γ_A = Σ_A∪Z∪ {y}, and Σ⁰ ⊆f in Σ such that Σ⁰ \Σ_A 6= ∅. We define on V R(Γ_A) the relation ≡Σ⁰

determined for everyh₁, h₂∈V R(Γ_A) by

h₁≡_Σ⁰ h₂ iff h₁(σ)∩Σ⁰=h₂(σ)∩Σ⁰ for everyσ∈Z∪ {y}.

It should be clear that ≡_Σ⁰ is an equivalence relation. Moreover, since Z ∪ {y}

and Σ⁰ are finite, the index of ≡_Σ⁰ is finite. LetV be a set of representatives of V R(ΓA)/≡Σ⁰. For everyh∈V, we letZh={z∈Z|h(z)∈Σ⁰}and Γh= ΣA∪ Σ⁰∪(Z\Zh)∪{y}, and we consider the weighted automatonAh= (Qh, inh, wth, terh) over Γh and K, where Qh = {qh|q∈Q} is a copy of Q, inh(qh) = in(q) and terh(qh) = ter(q) for every qh ∈ Qh. The weight assignment mapping wth is defined as follows. For everyqh, q⁰_h∈Qh, σ∈Γh, we let

wt_h((q_h, σ, q_h⁰)) =











wt((q, σ, q⁰)) ifσ∈Σ_A∪(Z\Z_h)∪ {y}

wt((q, z, q⁰)) ifσ=h(z) andz∈Zh

wt((q, y, q⁰)) ifσ∈h(y)∩Σ⁰

0 otherwise

.

Without any loss, we assume that the setsQh are pairwise disjoint. We letQV = [

h∈V

Qh, ΓV = ΣA ∪Σ⁰ ∪Z ∪ {y}, and consider the wva A_(Σ0,V)=

Σ, A_(Σ0,V)

over Σ and K, where A_(Σ⁰_,V) = (QV, inV, wtV, terV) is a weighted automaton with input alphabet ΓV. Its initial and final distribution are defined, respectively, by inV (q) = inh(q), terV (q) = terh(q) for every q ∈ Qh, h ∈ V. The weight assignment mappingwtV :QV ×ΓV ×QV →K is given by

wtV((q, σ, q⁰)) =

wt_h((q, σ, q⁰)) ifq, q⁰ ∈Q_hfor some h∈V

0 otherwise

for everyq, q⁰ ∈Q_V, σ∈Γ_V.

Since the weighted automaton A(Σ⁰,V) is the disjoint union of Ah, h∈ V, we get that

A_(Σ0,V)

=X

h∈V

kAhk. Therefore, for everyw∈Σ^∗, we have

A_(Σ0,V)

, w

= X

u∈preimΓV(w)

A_(Σ0,V)

, u

=X

h∈V

X

u∈preimΓh(w)

(kA_hk, u).

Lemma 1. kAk=

A_(Σ⁰_,V) .

Proof. Letw=w(0). . . w(n−1)∈Σ^∗. Consider a word u=u(0). . . u(n−1)∈ preimΓ_A(w) and a valid relabeling h ∈ V R(ΓA) with w ∈ h(u). We define the wordu⁰ =u⁰(0). . . u⁰(n−1)∈Γ^∗_V as follows:

u⁰(i) =

u(i) if (u(i)∈ΣA∪Z\Zh) or (u(i) =y andw(i)∈/ Σ⁰\ΣA) w(i) if (u(i)∈Z_h) or (u(i) =yand w(i)∈Σ⁰\Σ_A)

for every 0≤i≤n−1.

We consider the set of valid relabelings V⁰⊆V as follows: g∈V⁰ implies that g(z) = h(z) for every z ∈Zh∩ {u(i)|0≤i≤n−1} and g(y)∩Σ⁰ =h(y)∩Σ⁰ whenever u(i) =y and w(i)∈ Σ⁰ for some 0 ≤ i ≤ n−1. Let Pu^(A) be a path of A over u. Then, by construction of A_(Σ0,V), for every g ∈ V⁰, there exists a pathP_u^(A0^g)ofA_goveru⁰ withweight

P_u^(A0 ^g)

=weight Pu^(A)

. Clearly, there are r=card(V⁰) such paths and sinceK is idempotent, we get X

g∈V⁰

weight P_u^(A0 ^g)

= weight

Pu^(A)

. On the other hand, for everyg∈V \V⁰ and pathP_u^(A0 ^g)ofAg, we haveweight

P_u^(A0 ^g)

= 0. Therefore, we obtain X

P_u^(A)

weight P_u^(A)

=X

g∈V

X

P(Ag)

u0

weight P_u^(A0 ^g)

.

We define the valid relabelingh⁰∈V R(ΓV) as follows:

- h⁰(z) =h(z) for everyz∈Z\Zh, and we let, nondeterministically,

- h⁰(z) ∈Σ\(ΣA∪Σ⁰∪h(Z\Zh)∪ {w(i)|0≤i≤n−1 andw(i)∈h(y)}) for every z∈Zh.

Then we havew∈h⁰(u⁰) which implies thatu⁰∈preimΓ_V (w).

Conversely, let u⁰ =u⁰(0). . . u⁰(n−1) ∈preimΓ_V(w). Hence, there is a valid relabelingh⁰∈V R(ΓV) such thatw∈h⁰(u⁰). By construction ofA(Σ⁰,V), there is a valid relabelinghfrom Γ_A to Σ and a wordu=u(0). . . u(n−1)∈Γ^∗_A such that

u(i) =







u⁰(i) ifu⁰(i)∈ΣA∪Z\Zh

z ifu⁰(i) =h(z) andz∈Zh

y ifu⁰(i)∈(h(y)∩Σ⁰)∪ {y}

for every 0 ≤ i ≤ n−1. Keeping the previous notations, for every g ∈ V⁰, there is a path P_u^(A0 ^g) of the weighted automaton Ag over u⁰. By construction of A_(Σ⁰_,V), all such paths P_u^(A0 ^g) (g ∈ V⁰) have the same weight and there exist r = card(V⁰) such paths. Furthermore, for every g ∈ V⁰ and P_u^(A0^g) there is a path Pu^(A) of A over u with weight

Pu^(A)

= weight P_u^(A0 ^g)

, and since K is idempotent we getweight

Pu^(A)

= X

g∈V⁰

weight P_u^(A0 ^g)

. On the other hand, for everyg∈V\V⁰and pathP_u^(A0 ^g)ofAg, we have thatweight

P_u^(A0 ^g)

= 0. Therefore X

g∈V

X

P(Ag)

u0

weight P_u^(A0^g)

= X

P_u^(A)

weight Pu^(A)

. We consider the relabeling h⁰⁰

from Γ_A to Σ defined in the following way. It is the identity on Σ_A, h⁰⁰(z) = h⁰(z) for every z ∈ Z\Zh, h⁰⁰(z) = h(z) for every z ∈ Zh, and h⁰⁰(y) = h⁰(y)∪ ((h(y)∩Σ⁰)\h(Zh)) (in fact (h(y)∩Σ⁰)∩h(Zh) =∅sincehis a valid relabeling on ΓA). Triviallyh⁰⁰ is a valid relabeling and w ∈ h⁰⁰(u) which implies that u∈ preimΓ_A(w).

We conclude that for everyw∈Σ^∗ we have A_(Σ⁰_,V)

, w

= X

u⁰∈preimΓV(w)

A_(Σ⁰_,V) , u⁰

= X

u⁰∈preimΓV(w)

X

g∈V

(kAgk, u⁰)

= X

u⁰∈preimΓV(w)

X

g∈V

X

P(Ag)

u0

weight P_u^(A0^g)

= X

u∈preim_ΓA(w)

X

P_u^(A)

weight P_u^(A)

= X

u∈preimΓA(w)

(kAk, u) = (kAk, w) and we are done.

4 Closure properties of the class V Rec (K, Σ)

In this section, we investigate closure properties of the class of v-recognizable series over the infinite alphabet Σ and the semiringK. More precisely, we show that the classV Rec(K,Σ) is closed under sum, and under scalar, Hadamard, Cauchy and shuffle products, as well as star operation.

Proposition 4. The classV Rec(K,Σ)is closed under sum.

Proof. Let r⁽ⁱ⁾ ∈ V Rec(K,Σ) with i = 1,2. Then there exist two wva A⁽ⁱ⁾ = Σ, A⁽ⁱ⁾

with A⁽ⁱ⁾ = Q⁽ⁱ⁾, in⁽ⁱ⁾, wt⁽ⁱ⁾, ter⁽ⁱ⁾

and Γ⁽ⁱ⁾ = Σ⁽ⁱ⁾∪Z⁽ⁱ⁾∪

y⁽ⁱ⁾ , ac- cepting r⁽ⁱ⁾, for i = 1,2. Without any loss, we assume that Q⁽¹⁾ ∩Q⁽²⁾ = ∅ and Z⁽¹⁾∪

y⁽¹⁾ ∩ Z⁽²⁾∪

y⁽²⁾ = ∅. We consider the wva A⁽¹⁾

(^Σ(2),V1) =

Σ, A⁽¹⁾ (^Σ(2),V1)

withA⁽¹⁾

(^Σ(2),V1)= Q⁽¹⁾_V

1, in⁽¹⁾_V

1, wt⁽¹⁾_V

1, ter_V⁽¹⁾

1

over Γ⁽¹⁾∪Σ⁽²⁾ andK and the wvaA⁽²⁾

(^Σ(1),V2)=

Σ, A⁽²⁾ (^Σ(1),V2)

withA⁽²⁾

(^Σ(1),V2)= Q⁽²⁾_V

2, in⁽²⁾_V

2, wt⁽²⁾_V

2, ter⁽²⁾_V

2

over Γ⁽²⁾ ∪Σ⁽¹⁾ and K, determined by the procedure before Lemma 1. More- over, without any loss, we assume that Q⁽¹⁾_V

1 ∩Q⁽²⁾_V

2 = ∅. By Lemma 1 we have

A⁽¹⁾ (^Σ(2),V1)

= r⁽¹⁾ and

A⁽²⁾ (^Σ(1),V2)

= r⁽²⁾. Let Q = Q⁽¹⁾_V

1 ∪Q⁽²⁾_V

2 and Γ = Σ⁽¹⁾∪Σ⁽²⁾∪Z⁽¹⁾∪Z⁽²⁾∪ {y}, wherey denotes a new free variable different from y⁽¹⁾ andy⁽²⁾. We consider the wvaA=hΣ, AiwithA= (Q, in, wt, ter) where in andterare defined, for everyq∈Q, respectively by

in(q) =

( in⁽¹⁾_V

1 (q) ifq∈Q⁽¹⁾_V

1

in⁽²⁾_V

2 (q) ifq∈Q⁽²⁾_V

2

and ter(q) =

( ter_V⁽¹⁾

1 (q) ifq∈Q⁽¹⁾_V

1

ter_V⁽²⁾

2 (q) ifq∈Q⁽²⁾_V

2

.

The weight assignment mappingwt:Q×Γ×Q→K is defined as follows:

wt((q, σ, q⁰)) =

















 wt⁽¹⁾_V

1 ((q, σ, q⁰)) ifq, q⁰ ∈Q⁽¹⁾_V

1, σ∈Γ\ {y}

wt⁽²⁾_V

2 ((q, σ, q⁰)) ifq, q⁰ ∈Q⁽²⁾_V

2, σ∈Γ\ {y}

wt⁽¹⁾_V

1 q, y⁽¹⁾, q⁰

ifq, q⁰ ∈Q⁽¹⁾_V

1, σ=y wt⁽²⁾_V

2 q, y⁽²⁾, q⁰

ifq, q⁰ ∈Q⁽²⁾_V

2, σ=y

0 otherwise

for everyq, q⁰ ∈Q, σ∈Γ.

We show that kAk= A⁽¹⁾

+ A⁽²⁾

. For this, letw∈Σ^∗, u∈preim_Γ(w), and h ∈ V R(Γ) such that w ∈ h(u). Then, for every path Pu^(A) of A over u, by construction of A, we point out the following cases. (i) There exists a path P_u(1) of A⁽¹⁾

(^Σ(2),V1) over u⁽¹⁾ with weight(P_u(1)) = weight Pu^(A)

, where u⁽¹⁾ is obtained fromu by replacing every occurrence ofy with y⁽¹⁾. (ii) There exists a pathP_u(2) of A⁽²⁾

(^Σ(1),V2)over u⁽²⁾ with weight(P_u(2)) =weight Pu^(A)

, whereu⁽²⁾

is obtained fromuby replacing every occurrence ofywithy⁽²⁾. Suppose firstly that (i) holds. We consider the valid relabeling h⁽¹⁾ ∈V R Γ⁽¹⁾∪Σ⁽²⁾

such thath⁽¹⁾ coincides withhon Σ⁽¹⁾∪Σ⁽²⁾∪Z⁽¹⁾ andh⁽¹⁾ y⁽¹⁾

=h(y)∪h Z⁽²⁾

. Trivially, w∈h⁽¹⁾ u⁽¹⁾

which implies that u⁽¹⁾ ∈preim_Γ(1)∪Σ⁽²⁾(w). Similarly, in case (ii) we get thatu⁽²⁾∈preim_Γ(2)∪Σ⁽¹⁾(w).

Conversely, let w∈Σ^∗, u⁽¹⁾ ∈preim_Γ(1)∪Σ⁽²⁾(w), andh⁽¹⁾ ∈V R Γ⁽¹⁾∪Σ⁽²⁾ such that w ∈ h⁽¹⁾ u⁽¹⁾

. Then, for every path P_u(1) of A⁽¹⁾

(^Σ(2),V1) over u⁽¹⁾, by construction of A, there exists a path Pu^(A) of A over u with weight

Pu^(A)

= weight(P_u(1)), where uis obtained fromu⁽¹⁾ by replacing every occurrence ofy⁽¹⁾ with y. We define the valid relabeling h ∈ V R(Γ) which coincides withh⁽¹⁾ on Σ⁽¹⁾∪Σ⁽²⁾∪Z⁽¹⁾,h(z)∈Σ\(Σ⁽¹⁾∪Σ⁽²⁾∪h⁽¹⁾ Z⁽¹⁾

∪(h⁽¹⁾ y⁽¹⁾

∩{w(i)|0 ≤i≤n−1})) for z ∈Z⁽²⁾ and h(y) =h⁽¹⁾ y⁽¹⁾

\ {h(z)|z ∈Z⁽²⁾}.

Trivially,w∈h(u) which implies thatu∈preim_Γ(w).

Next assume thatu⁽²⁾ ∈preim_Γ(2)∪Σ⁽¹⁾(w) and h⁽²⁾ ∈V R Γ⁽²⁾∪Σ⁽¹⁾

such that w∈h⁽²⁾ u⁽²⁾

. Then, for every pathP_u(2) ofA⁽²⁾

(^Σ(1),V2)overu⁽²⁾, by construction of A, there exists a pathP_u^A0 ofAoveru⁰ withweight

P_u^(A)0

=weight(P_u(2)), where u⁰ is obtained from u⁽²⁾ by replacing every occurrence of y⁽²⁾ with y. We define the valid relabeling h⁰ ∈ V R(Γ) which coincides with h⁽²⁾ on Σ⁽¹⁾∪Σ⁽²⁾∪Z⁽²⁾, h⁰(z) ∈ Σ\ Σ⁽¹⁾∪Σ⁽²⁾∪h⁽²⁾ Z⁽²⁾

∪ h⁽²⁾ y⁽²⁾

∩ {w(i)|0≤i≤n−1}

for z ∈Z⁽¹⁾ and h⁰(y) =h⁽²⁾ y⁽²⁾

\ {h⁰(z) |z ∈Z⁽¹⁾}. Trivially, w ∈h⁰(u⁰) which implies thatu⁰ ∈preimΓ(w).

We conclude that for everyw∈Σ^∗ we have (kAk, w) = X

u∈preim_Γ(w)

(kAk, u) = X

u∈preim_Γ(w)

X

P_u^(A)

weight P_u^(A)

= X

u⁽¹⁾∈preim_{Γ(1)∪Σ(2)}(w)

X

Pu(1)

weight(P_u(1))

+ X

u⁽²⁾∈preim_{Γ(2)∪Σ(1)}(w)

X

Pu(2)

weight(P_u(2))

= X

u⁽¹⁾∈preim_{Γ(1)∪Σ(2)}(w)

A⁽¹⁾ (^Σ(2),V1)

, u⁽¹⁾

+ X

u⁽²⁾∈preim_{Γ(2)∪Σ(1)}(w)

A⁽²⁾ (^Σ(1),V2)

, u⁽²⁾

=

A⁽¹⁾ (^Σ(2),V1)

, w

+

A⁽²⁾ (^Σ(1),V2)

, w

=

A⁽¹⁾

, w

+

A⁽²⁾

, w

=

A⁽¹⁾

+

A⁽²⁾

, w

=

r⁽¹⁾+r⁽²⁾, w

where the sixth equality holds by Lemma 1, and we are done.

Proposition 5. The classV Rec(K,Σ)is closed under the scalar products.

Proof. Let r ∈ V Rec(K,Σ) and k ∈ K. Then there exists a wva A = hΣ, Ai with A = (Q, in, wt, ter) accepting r. We consider the wva A⁰ = hΣ, A⁰i with A⁰ = (Q, in⁰, wt, ter) wherein⁰(q) =k·in(q) for everyq∈Q. Then, by standard arguments we getkA⁰k=kkAk, and we are done.

Proposition 6. The classV Rec(K,Σ)is closed under Hadamard product.

Proof. Let r⁽ⁱ⁾ ∈ V Rec(K,Σ) with i = 1,2. Then there exist two wva A⁽ⁱ⁾ = Σ, A⁽ⁱ⁾

with A⁽ⁱ⁾ = Q⁽ⁱ⁾, in⁽ⁱ⁾, wt⁽ⁱ⁾, ter⁽ⁱ⁾

over Γ⁽ⁱ⁾ = Σ⁽ⁱ⁾ ∪Z⁽ⁱ⁾∪ y⁽ⁱ⁾ , accepting r⁽ⁱ⁾ for i = 1,2. Without any loss, we assume that Q⁽¹⁾∩Q⁽²⁾ = ∅ and Z⁽¹⁾∪

y⁽¹⁾ ∩ Z⁽²⁾∪

y⁽²⁾ = ∅. We consider the wva A⁽¹⁾

(^Σ(2),V1) =

Σ, A⁽¹⁾ (^Σ(2),V1)

with A⁽¹⁾

(^Σ(2),V1) = Q⁽¹⁾_V

1, in⁽¹⁾_V

1, wt⁽¹⁾_V

1, ter_V⁽¹⁾

1

over Γ⁽¹⁾∪Σ⁽²⁾ and A⁽²⁾

(^Σ(1),V2)=

Σ, A⁽²⁾ (^Σ(1),V2)

withA⁽²⁾

(^Σ(1),V2)= Q⁽²⁾_V

2, in⁽²⁾_V

2, wt⁽²⁾_V

2, ter_V⁽²⁾

2

over Γ⁽²⁾∪ Σ⁽¹⁾ determined by the procedure described before Lemma 1. Moreover, without any loss, we assume thatQ⁽¹⁾_V

1 ∩Q⁽²⁾_V

2 =∅. By Lemma 1 we get

A⁽¹⁾ (^Σ(2),V1)

=r⁽¹⁾ and

A⁽²⁾ (^Σ(1),V2)

=r⁽²⁾.

We consider the set Z⁽¹⁾∪

y⁽¹⁾ × Z⁽²⁾∪

y⁽²⁾ \{y}wherey= y⁽¹⁾, y⁽²⁾ , and a maximal subset G ⊆ Z⁽¹⁾∪

y⁽¹⁾ × Z⁽²⁾∪

y⁽²⁾ \ {y} satisfying the next condition: every element ofZ⁽¹⁾(resp. ofZ⁽²⁾) occurs in at most one pair inG as a left (resp. as a right) coordinate. Assume thatG1, . . . , Gmis an enumeration of all such sets of pairs of variables. Moreover, we letQ=Q⁽¹⁾_V

1 ×Q⁽²⁾_V

2 and ΓG_j = Σ⁽¹⁾∪Σ⁽²⁾∪Gj ∪ {y} for every 1 ≤ j ≤ m, and we consider the wva AGj = Σ, A_Gj

with A_Gj = Q, in_Gj, wt_Gj, ter_Gj

over Γ_Gj. For every 1 ≤ j ≤ m, the initial and terminal distribution are given respectively, byin_Gj q⁽¹⁾, q⁽²⁾

= in⁽¹⁾_V

1 q⁽¹⁾

·in⁽²⁾_V

2 q⁽²⁾

andterG_j q⁽¹⁾, q⁽²⁾

=ter⁽¹⁾_V

1 q⁽¹⁾

·ter⁽²⁾_V

2 q⁽²⁾

, and the weight assignment mappingwtG_j :Q×ΓG_j ×Q→Kis defined by

wtG_j

q⁽¹⁾, q⁽²⁾ , σ,

q⁰⁽¹⁾, q⁰⁽²⁾

=





 wt⁽¹⁾_V

1 q⁽¹⁾, σ, q⁰⁽¹⁾

·wt⁽²⁾_V

2 q⁽²⁾, σ, q⁰⁽²⁾

ifσ∈Σ⁽¹⁾∪Σ⁽²⁾ wt⁽¹⁾_V

1 q⁽¹⁾, x⁽¹⁾, q⁰⁽¹⁾

·wt⁽²⁾_V

2 q⁽²⁾, x⁽²⁾, q⁰⁽²⁾

ifσ= x⁽¹⁾, x⁽²⁾

∈G_j∪ {y}

0 otherwise

for every q⁽¹⁾, q⁽²⁾

, q⁰⁽¹⁾, q⁰⁽²⁾

∈Q, σ∈ΓG_j.

By Proposition 4, the series X

1≤j≤m

AG_j

is recognizable. We will show that A⁽¹⁾

A⁽²⁾

= X

1≤j≤m

A_Gj .

To this end, let w=w(0). . . w(n−1) ∈Σ^∗, u⁽¹⁾ ∈preim_Γ(1)∪Σ⁽²⁾(w),u⁽²⁾ ∈ preim_Γ(2)∪Σ⁽¹⁾(w),h⁽¹⁾∈V R Γ⁽¹⁾∪Σ⁽²⁾

, andh⁽²⁾∈V R Γ⁽²⁾∪Σ⁽¹⁾

such that w ∈h⁽¹⁾ u⁽¹⁾

∩h⁽²⁾ u⁽²⁾

. For every w(t)∈ Σ, 0≤ t ≤n−1, we have either w(t)∈ Σ⁽¹⁾∪Σ⁽²⁾ and hence u⁽¹⁾(t) =u⁽²⁾(t) =w(t), or w(t)∈Σ\Σ⁽¹⁾∪Σ⁽²⁾ and one of the following cases holds.

• There exist bounded variablesz⁽¹⁾∈Z⁽¹⁾,z⁽²⁾∈Z⁽²⁾such thatu⁽¹⁾(t) =z⁽¹⁾, u⁽²⁾(t) =z⁽²⁾ andh⁽¹⁾ u⁽¹⁾(t)

=h⁽²⁾ u⁽²⁾(t)

=w(t).

• There exists a bounded variablez⁽¹⁾∈Z⁽¹⁾such thatu⁽¹⁾(t) =z⁽¹⁾,u⁽²⁾(t) = y⁽²⁾ andh⁽¹⁾ u⁽¹⁾(t)

=w(t)∈h⁽²⁾ u⁽²⁾(t) .

• There exists a bounded variablez⁽²⁾ ∈Z⁽²⁾such thatu⁽¹⁾(t) =y⁽¹⁾,u⁽²⁾(t) = z⁽²⁾ andh⁽²⁾ u⁽²⁾(t)

=w(t)∈h⁽¹⁾ u⁽¹⁾(t) .

• u⁽¹⁾(t) =y⁽¹⁾,u⁽²⁾(t) =y⁽²⁾, andw(t)∈h⁽¹⁾ u⁽¹⁾(t)

∩h⁽²⁾ u⁽²⁾(t) . We consider the wordu=u(0). . . u(n−1) by

u(t) =

w(t) ifw(t)∈Σ⁽¹⁾∪Σ⁽²⁾ u⁽¹⁾(t), u⁽²⁾(t)

otherwise

for every 0 ≤ t ≤ n−1. For every 1 ≤ j ≤ m, we define a valid relabel- ing hj ∈ V R ΓG_j

such that hj(σ) = h⁽¹⁾ x⁽¹⁾

for every σ = x⁽¹⁾, x⁽²⁾

∈ Z⁽¹⁾× Z⁽²⁾∪

y⁽²⁾ ∩Gj, and hj(σ) =h⁽²⁾ x⁽²⁾

for everyσ= x⁽¹⁾, x⁽²⁾

∈ y⁽¹⁾ ×Z⁽²⁾

∩Gj. Henceu∈preimΓ_Gj (w) for some 1≤j ≤m.

By the definition of the listG₁, . . . , G_m, there is a setJ ⊆ {1, . . . , m}, such that for every path

P_u(1) :

q⁽¹⁾₀ , u⁽¹⁾(0), q⁽¹⁾₁ . . .

q_n−1⁽¹⁾ , u⁽¹⁾(n−1), q⁽¹⁾n

ofA⁽¹⁾

(^Σ(2),V1)overu⁽¹⁾, and P_u(2) :

q⁽²⁾₀ , u⁽²⁾(0), q⁽²⁾₁ . . .

q_n−1⁽²⁾ , u⁽²⁾(n−1), q⁽²⁾n

ofA⁽²⁾

(^Σ(1),V2)overu⁽²⁾, there exists a path Pu^(Gj):

q⁽¹⁾₀ , q⁽²⁾₀

, u(0),

q₁⁽¹⁾, q₁⁽²⁾ . . .

q⁽¹⁾_n−1, q_n−1⁽²⁾

, u(n−1),

q⁽¹⁾n , q⁽²⁾n

ofA_Gj overu, for everyj∈J. Conversely, for every pathPu^(Gj)ofA_Gj overu(j∈ J) there are two pathsP_u(1) ofA⁽¹⁾

(^Σ(2),V1)overu⁽¹⁾ andP_u(2) ofA⁽²⁾

(^Σ(1),V2)overu⁽²⁾ respectively, obtained in the obvious way. Moreover, in caseweight

Pu^(Gj)

6= 0, for everyj∈J, it holds

weight P_u^(Gj)

=inG_j

q⁽¹⁾₀ , q₀⁽²⁾

· Y

0≤t≤n−1

wtG_j

q⁽¹⁾_t , q_t⁽²⁾

, u(t),

q⁽¹⁾_t+1, q⁽²⁾_t+1

·ter_Gj

q⁽¹⁾_n , q⁽²⁾_n

=in⁽¹⁾_V

1

q₀⁽¹⁾

·in⁽²⁾_V

2

q⁽²⁾₀

· Y

0≤t≤n−1



 wt⁽¹⁾_V

1

q_t⁽¹⁾, u⁽¹⁾(t), q_t+1⁽¹⁾

·wt⁽²⁾_V

2

q_t⁽²⁾, u⁽²⁾(t), q⁽²⁾_t+1





·ter_V⁽¹⁾

1

q⁽¹⁾_n

·ter_V⁽²⁾

2

q⁽²⁾_n

=in⁽¹⁾_V

1

q₀⁽¹⁾

· Y

0≤t≤n−1

wt⁽¹⁾_V

1

q⁽¹⁾_t , u⁽¹⁾(t), q⁽¹⁾_t+1

·ter⁽¹⁾_V

1

q_n⁽¹⁾

·in⁽²⁾_V

2

q₀⁽²⁾

· Y

0≤t≤n−1

wt⁽²⁾_V

2

q⁽²⁾_t , u⁽²⁾(t), q_t+1⁽²⁾

·ter_V⁽²⁾

2

q⁽²⁾_n

=weight(P_u(1))·weight(P_u(2)).

Conversely, if weight(P_u(1)) 6= 0, weight(P_u(2)) 6= 0, then by the consideration of the list G1, . . . , Gm, there is at least one 1 ≤ j ≤ m with weight

Pu^(Gj)

= weight(P_u(1))·weight(P_u(2)). Therefore, and since Kis idempotent, we obtain²

X

1≤j≤m

X

P(Gj)

u

weight P_u^(Gj)

= X

Pu(1),P

u(2)

weight(P_u(1))·weight(P_u(2)).

We conclude



 X

1≤j≤m

AG_j

, w



= X

1≤j≤m

AG_j

, w

= X

1≤j≤m

X

u∈preimΓGj(w)

AG_j

, u

= X

1≤j≤m

X

u∈preimΓGj(w)

X

P(Gj)

u

weight P_u^(Gj)

= X

u⁽¹⁾∈preim_{Γ(1)∪Σ(2)}(w) u⁽²⁾∈preim_{Γ(2)∪Σ(1)}(w)

X

Pu(1)

Pu(2)

(weight(P_u(1))·weight(P_u(2)))

= X

u⁽¹⁾∈preim_{Γ(1)∪Σ(2)}(w)

X

Pu(1)

weight(P_u(1))

2It should be clear that forj∈ {1, . . . , m} \Jthe pathsPu^(Gj)do not exist, hence by definition weight

Pu^(Gj)

= 0.